Complex Arithmetic/Examples/(5 + 5i) (3 - 4i)^-1 + 20 (4 + 3i)^-1

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Example of Complex Arithmetic

$\dfrac {5 + 5 i} {3 - 4 i} + \dfrac {20} {4 + 3 i} = 3 - i$


Proof

\(\ds \dfrac {5 + 5 i} {3 - 4 i}\) \(=\) \(\ds \dfrac {\paren {5 + 5 i} \paren {3 + 4 i} } {\paren {3 - 4 i} \paren {3 + 4 i} }\) multiplying top and bottom by $3 + 4 i$
\(\ds \) \(=\) \(\ds \dfrac {15 + 15 i + 20 i + 20 i^2} {3^2 + 4^2}\) Definition of Complex Multiplication
\(\ds \) \(=\) \(\ds \dfrac {-5 + 35 i} {25}\) simplifying


Then:

\(\ds \dfrac {20} {4 + 3 i}\) \(=\) \(\ds \dfrac {20 \paren {4 - 3 i} } {\paren {4 + 3 i} \paren {4 - 3 i} }\) multiplying top and bottom by $4 - 3 i$
\(\ds \) \(=\) \(\ds \dfrac {80 - 60 i} {4^2 + 3^2}\) Definition of Complex Multiplication
\(\ds \) \(=\) \(\ds \dfrac {80 - 60 i} {25}\) simplifying


Finally:

\(\ds \dfrac {-5 + 35 i} {25} + \dfrac {80 - 60 i} {25}\) \(=\) \(\ds \dfrac {\paren {-5 + 80} + \paren {35 - 60} i} {25}\) Definition of Complex Addition
\(\ds \) \(=\) \(\ds \dfrac {75 - 25 i} {25}\)
\(\ds \) \(=\) \(\ds 3 - i\) simplifying

$\blacksquare$


Sources